Problem: For the graph of a certain quadratic $y = ax^2 + bx + c$, the vertex of the parabola is $(3,7)$ and one of the $x$-intercepts is $(-2,0)$.  What is the $x$-coordinate of the other $x$-intercept?
Answer: Since the vertex of the parabola is $(3,7)$, the parabola is symmetric around the line $x = 3$.  Furthermore, the two $x$-intercepts of the parabola are also symmetric around this line.  One $x$-intercept is $(-2,0)$, whose distance from the line $x = 3$ is $3 - (-2) = 5$, so the other $x$-intercept is at $(3 + 5,0) = (8,0)$.  The $x$-coordinate of this $x$-intercept is $\boxed{8}$.